Question
$$\frac{x}{x+1}+\frac{x+1}{x} = 0$$
Answer
$$ \begin{matrix}x_1 = -\dfrac{ 1 }{ 2 }+\dfrac{ 1 }{ 2 }i & x_2 = -\dfrac{ 1 }{ 2 }-\dfrac{ 1 }{ 2 }i \\[1 em] \end{matrix} $$
Explanation
$$ \begin{aligned} \frac{x}{x+1}+\frac{x+1}{x} &= 0&& \text{multiply ALL terms by } \color{blue}{ (x+1)x }. \\[1 em](x+1)x \cdot \frac{x}{x+1}+(x+1)x\frac{x+1}{x} &= (x+1)x\cdot0&& \text{cancel out the denominators} \\[1 em]x^2+x^2+2x+1 &= 0&& \text{simplify left side} \\[1 em]2x^2+2x+1 &= 0&& \\[1 em] \end{aligned} $$
$ 2x^{2}+2x+1 = 0 $ is a quadratic equation.
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